The legacy of 40 years of cuprate superconductivity

In February 1986, Bednorz and Müller made a stunning discovery: superconductivity at a temperature of 35 K in a doped copper oxide (cuprate). Arguably, this discovery changed condensed matter physics. In April 1986, they submitted their results to Z. Phys. B. Only nineteen months later, they were awarded the Nobel Prize in Physics, the shortest time ever between a discovery and the award. A nice and short review of the history is here.

One measure of my estimate of the influence of this discovery is that it received about 5 pages of coverage in my Condensed Matter Physics: A Very Short Introduction. (See Chapter 5, Adventures in Flatland).

How things have developed over the past forty years, for better and worse, may be representative of how science advances: discovery by serendipity, hype about applications, unexpected secondary benefits, foundational questions, new concepts, unification, and incremental advances.

Hype about technological applications

On March 20, 1987, The New York Times had a front-page article, DISCOVERIES BRING A 'WOODSTOCK' FOR PHYSICS, by James Gleick. This followed the 1987 APS March meeting. It began 

"Physicists from three continents converged on the New York Hilton for a hastily scheduled special conference on a string of discoveries that seem certain to produce a rapid cascade of commercial applications in electricity, magnetism and electronics.There are many things we know and understand that we did not when they were first discovered."

This has largely been unfulfilled. There are a few niche applications, but cuprates are not used in electricity distribution or even in the superconducting magnets in hospital MRI machines, which are probably the main commercial application of superconductors. One of the significant obstacles is that it is hard to make wires from these materials, as they are ceramics. This is an example of the common gap between research laboratory science and commercially viable technology.

After 40 years, do we have a successful theory?

It depends on who you ask. But I would say there is a lot we do understand.

We have a phenomenological theory for all the macroscopic phenomena associated with the superconducting state: Ginzburg-Landau theory!

Properties of the superconducting state are well-described by a BCS wavefunction with a d-wave order parameter and the associated Bogoliubov quasiparticles. [This is somewhat puzzling, as in the metallic state quasi-particles are not well defined].

Although not everyone agrees, I think it is fair to say that the essential physics is in a one-band Hubbard model, and the key physics is:

strong electronic correlations,

a doped antiferromagnetic Mott insulator,

d-wave pairing that is "mediated"/caused from some mixture/variant of antiferromagnetic spin fluctuations or RVB spin singlets,.....

We certainly don't understand the cuprates at the same level as elemental superconductors. But we do understand the essential physics.

What is harder to describe and understand are the states adjacent to the superconducting state in the phase diagram: the pseudogap state and the strange metal.

Strongly correlated electron materials became a large, vibrant and unified field

Before 1986, there were small, disconnected communities intermittently interested in transition metal oxides, rare earths, Kondo impurities, Mott metal-insulator transitions, organic superconductors, heavy fermions, and quantum antiferromagnets.

The discovery of the cuprates brought together these communities as they found common interests, challenges, questions, concepts, and techniques.

The discovery of superconductivity in strontium ruthenate, alkali fullerides, iron pnictides and chalcogenides, twisted bilayer graphene and more cuprates, organic charge-transfer salts, and heavy fermions has shown how rich these systems are. The challenge is to understand the similarities and differences between these chemically and structurally diverse systems. In many of them, superconductivity is proximate to a Mott insulating state.

The unity and excitement were probably stimulated and enhanced by the activities and ideas of high-profile theorists such as Anderson, Schrieffer, Scalapino, Pines, Rice, and Varma. On the other hand, their acrimonious disagreements probably did not help.

Secondary theoretical benefits

The things I list below were not new ideas when the cuprate discovery happened. However, interest in the cuprates led them to become major research themes and ideas.

Importance of phase diagrams, including as a function of interaction parameters in toy models

Highlighting the limitations of electronic structure methods based on Density Functional Theory with approximate Exchange-Correlation functionals (i.e., anything computational). In the presence of strong correlations, DFT methods have spectacular failures. For example, predicting a metallic state instead of the Mott insulator.

Low dimensionality leads to qualitatively different behaviour, including the possibility of new types of order and quasiparticles. This is most dramatic in one dimension, where one has Luttinger liquids and spin-charge separation.

Spin liquids. Landau was wrong. Spontaneous symmetry breaking does not always occur in antiferromagnets.

Non-Fermi liquids. Landau was wrong. Not all metals are Fermi liquids.

Quantum criticality. Although this is a robust concept for certain toy models, whether it is relevant to the cuprates remains contentious.

Systematic improvements in approximation schemes and numerical techniques - exact diagonalisation, DMRG, DMFT, quantum Monte Carlo,...

Emergence. Chemical complexity and strong interactions can lead to new states of matter.

Secondary experimental benefits

Better probes. The desire to characterise the cuprates helped drive significant improvements in the resolution of ARPES (Angle-Resolved PhotoEmission Spectroscopy), STM (Scanning Tunnelling Microscopy), and inelastic neutron scattering. These advances have born fruit in the study of a wide range of other materials, beyond the cuprates.

Growth of single crystals. The early days of the cuprates produced a lot of junk experimental results because of the poor quality of the samples produced by "shake and bake". However, the involvement of solid-state chemists has improved things. The techniques have also led to the production of single crystals for a wide range of strongly correlated materials.

Why is there so little research on cuprates today?

Today, there is little research directly on cuprates, both theoretically and experimentally. It is hard to get funding to work on them, even though there is a lot we don't understand really well.

This is because of the problem of fashion in science. The low-lying fruit has been picked. There is a continuous new stream of materials being discovered with exotic properties, the latest being twisted bilayer van der Waals compounds.

Angle-Dependent Magnetoresistance as a probe of Fermi surface properties in cuprates

About twenty-five years ago I became interested in how the Fermi surface of the metallic state of organic charge-transfer salts could be mapped out by measuring the interlayer resistance as a function of the direction of a large applied magnetic field. [A nice review from 2004 is by Mark Kartsovnik]. Later this technique was used for a range of other metals including strontium ruthenate, iron pnictides, semiconductor heterostructures, and finally cuprates, mostly in the overdoped region.

For the cuprates, it was discovered that one could not only map out the shape of the intralayer Fermi surface, but also anisotropies in the scattering rate and the interlayer hopping integral. Of particular interest was the finding that the overdoped cuprates were not simple Fermi liquids, as usually claimed, but more like anisotropic marginal Fermi liquids.

It should be stressed that the Fermi surface information is extracted indirectly by comparing experimental curves of angle-dependence to calculations based on different models for the shape of the Fermi surface, anisotropies in the scattering rate, and interlayer hopping. Thus, there is a fair bit of curve fitting to determine the parameters of the model. However, when one has observations at several magnetic fields, temperatures, and curves for the angle dependence in all directions, there are a lot of constraints, and specific anisotropies tend to produce some specific qualitative features in the shapes of the curves. Examples are shown below, taken from the Nature paper referenced below.

Recently, measurements have been reported on samples of the cuprate Nd-LSCO 

[La1.6−xNd0.4SrxCuO4] at dopings of p=0.21 and p=0.24, lying on both sides of the putative quantum critical point at p=0.23. 

The differences between the ADMR at these two dopings are analysed quantitatively in a preprint, which claims to show that at p=0.21 the Fermi surface is reconstructed due to (pi,pi) ordering. This is important as it relates to the fundamental question as to the origin of the pseudogap state.

Fermi surface transformation at the pseudogap critical point of a cuprate superconductor

Yawen Fang, Gael Grissonnanche, Anaelle Legros, Simon Verret, Francis Laliberte, Clement Collignon, Amirreza Ataei, Maxime Dion, Jianshi Zhou, David Graf, M. J. Lawler, Paul Goddard, Louis Taillefer, B. J. Ramshaw

Submitted on 3 Apr 2020 (v1), last revised 26 Nov 2020 (v2)

Aside: There is also a Nature paper, Linear-in temperature resistivity from an isotropic Planckian scattering rate, by the same group that compares the p=0.24 observations to those on the overdoped cuprate Tl2201 [p=0..29]. The arxiv notes "substantial text overlap" between the preprint above and the preprint for the Nature paper. [Figure 2 in v1 of the preprint above is in the Nature paper].

Here I focus on the first preprint as it stimulated a nice theory preprint

Interpreting Angle Dependent Magnetoresistance in Layered Materials: Application to Cuprates

Seth Musser, Debanjan Chowdhury, Patrick A. Lee, T. Senthil

They present a strong case against the main claim of Fang et al. that their ADMR data supports a reconstructed Fermi surface for the p=0.21 system.

There are several nice things about this preprint.

1. It shows how one should be careful about interpreting ADMR

2. It highlights the possible role of an anisotropic quasi-particle weight, Z(phi), where phi denotes the position on the intralayer Fermi surface, not the direction of the field. Anisotropy can arise from correlation effects and or "coherence factors" associated with Fermi surface reconstruction due to an ordered state. 

2. In their modeling, Fang et al. did not include the effects of Z(phi) and Musser et al. show that when it is included the qualitative differences in the ADMR that they claim arise due to the ordered state do not appear.

3. The authors consider a "toy" model for which some analytical results can be obtained. 

4. This provides some physical insight into the origins of the different features in the data, such as the peak around theta=40 degrees [It is just the magic angle associated with the average radius of the Fermi surface] and how the behaviour near theta=90 degrees depends on the relative size of different parameters [see especially equation (16)].

5. What is happening in this material may not be generic to the cuprates. "The van Hove filling in Nd-LSCO is located between the two dopings, p = 0.21 and p = 0.24, respectively. Thus what was a large Fermi surface centered at the Γ-point on the overdoped side will become a Fermi surface centered at (π, π) on the underdoped side, assuming no reconstruction occurs"

6. The most important insight is at the beginning of Section V. When the value of of the interlayer hopping integral t_perp(phi) averaged over the Fermi surface, changes from non-zero to zero an upturn in the ADMR at low angles (i.e. fields almost parallel to the layers) to a downturn. This suggests an alternative explanation for the transition seen in the preprint.

7. It highlights the often overlooked fact that observation of ADMR is not conclusive evidence of a three-dimensional Fermi surface. Using the parameters from the experimental preprint gives typical values of t_perp * tau ~ 0.1, and so the materials are far from the regime of a coherent three-dimensional Fermi surface.

I have a few minor comments

a. Like many others, the authors incorrectly credit with Yamaji explaining the magic angles associated with ADMR. However, Yamaji's explanation is not the correct one because it involves quantised orbits, whereas the effect is semi-classical, as explained by Kartsovnik, Laukhin, Pesotskii, Schegolev, and Yakovenko. 

b. Investigation of the role of small closed orbits when the magnetic field is almost parallel to the layers is credited to Schofield and Cooper. However, there was earlier and more detailed work by Hanasaki et al. Albeit, both of these papers consider the clean high field limit and so are of debatable relevance.

c. It would be nice to know the status of Fang et al., preprint on which this paper is based, particularly as the first authors of both are in the same department.

A new picture of unconventional superconductivity

Two key ideas concerning unconventional superconductors are the following.

1. s-wave and p-wave pairing (in momentum space) are associated with spin singlet and spin triplet pairing, respectively. This can be shown with minimal assumptions (no spin-orbit coupling and spatial inversion symmetry).

2. If superconductivity is seen in proximity to an ordered phase (e.g. ferromagnetism or antiferromagnetism) with a quantum critical point (QCP) then the pairing can be "mediated" by low energy fluctuations (e.g. magnons) associated with the ordering.

3. Non-fermi liquid behaviour may be seen in the quantum critical region about the QCP.

However, an interesting paper shows that neither of the above is necessarily true.

Superconductivity from Emerging Magnetic Moments 
Shintaro Hoshino and Philipp Werner

They find spin triplet superconductivity with s-wave symmetry. This arises because there is more than one orbital per site and due to the Hund's rule coupling spin triplets can form on a single site.

They also find the pairing is strongest near the "spin freezing crossover" which is associated with the "Hund's metal", i.e. the bad metal arising from the Hund's rule interaction, and has certain "non-Fermi liquid" properties.

The results are summarised in the phase diagrams below, which has a striking similarity to various experimental phase diagrams that are usually interpreted in terms of 2. above.
However, all the theory is DMFT and so there are no long wavelength fluctuations.

A concrete example of a quantum critical metal

I welcome comments on this preprint.

Quantum critical local spin dynamics near the Mott metal-insulator transition in infinite dimensions Nagamalleswararao Dasari, N. S. Vidhyadhiraja, Mark Jarrell, and Ross H. McKenzie

Finding microscopic models for metallic states that exhibit quantum critical properties such as $\omega/T$ scaling is a major theoretical challenge. We calculate the local dynamical spin susceptibility $\chi(T,\omega)$ for a Hubbard model at half filling using Dynamical Mean-Field Theory, which is exact in infinite dimensions. Qualitatively distinct behavior is found in the different regions of the phase diagram: Mott insulator, Fermi liquid metal, bad metal, and a quantum critical region above the finite temperature critical point. The signature of the latter is $\omega/T$ scaling where $T$ is the temperature. Our results are consistent with previous results showing scaling of the dc electrical conductivity and are relevant to experiments on organic charge transfer salts.

Here is the omega/T scaling, which I think is quite impressive.

We welcome comments.

Visit to a state university in India

Like everything in India, higher education is incredibly diverse, both in quality, resources, and culture. These statistics give some of the flavour. There are about 800 universities. A significant distinction is between state and central universities. The former are funded and controlled by state governments. The latter (and IITs, IISERs, IISc, TIFR...)  are funded and controlled by the central (i.e. national/federal) government. Broadly, the quality, resources, and autonomy (i.e. freedom from political interference) of the latter is much greater. On my many trips to India I have only visited these centrally funded institutes and universities.

This afternoon I looking forward to visiting the Physics Department of Vidyasagar University. It is funded by the West Bengal state government, and was started in 1981. It is named in honour of Ishwar Chandra Vidyasagar, a significant social reformer from the 19th century.

I am giving my talk on "Emergent Quantum Matter".
Here are the slides.

Update. I enjoyed my visit and interacting with the faculty and students. On the positive side, people were enthusiastic and there were some excellent questions from the students. I want to write a blog post about one question. On the negative side, it is sad to see how poorly places like this are resourced: whether infrastructure, lab equipment, lab supplies, library, faculty, or salaries. For example, there are 5 physics faculty members and they teach a full M.Sc. [2 years course work] to about 100 students. This is 2 courses per faculty per semester and obviously, their expertise is stretched to cover all courses. The Ph.D. students mostly have full-time jobs elsewhere and come in the afternoons and evenings to work on their projects. One travels 2 hours each way on public transport.

A critical review of holographic claims about condensed matter

There is a very helpful review article
Demystifying the Holographic Mystique by Dmitri Khveshchenko

In order to motivate a proper full reading I just give a few choice quotes.

Thus far, however, a flurry of the traditionally detailed (hence, rarely concise) publications on the topic have generated not only a good deal of enthusiasm but some reservations as well. Indeed, the proposed ’ad hoc’ generalizations of the original string-theoretical construction involve some of its most radical alterations, whereby most of its stringent constraints would have been aban- doned in the hope of still capturing some key aspects of the underlying correspondence. This is because the target (condensed matter) systems generically tend to be neither conformally, nor Lorentz (or even translationally and/or rotationally) invariant and lack any supersymmetric (or even an ordinary) gauge symmetry with some (let alone, large) rank-N non-abelian group. 

Moreover, while sporting a truly impressive level of technical profess, the exploratory ’bottom-up’ holographic studies have not yet helped to resolve such crucially important issues as: 

• Are the conditions of a large N, (super)gauge sym- metry, Lorentz/translational/rotational invariance of the boundary (quantum) theory indeed necessary for establishing a holographic correspondence with some weakly coupled (classical) gravity in the bulk? 

• Are all the strongly correlated systems (or only a precious few) supposed to have gravity duals? 

• What are the gravity duals of the already documented NFLs? 

• Given all the differences between the typical condensed matter and string theory problems, what (other than the lack of a better alternative) justifies the adaptation ’ad verbatim’ of the original (string-theoretical) holographic ’dictionary’? 

and, most importantly: 

• If the broadly defined holographic conjecture is indeed valid, then why is it so? 

Considering that by now the field of CMT holography has grown almost a decade old, it would seem that answering such outstanding questions should have been considered more important than continuing to apply the formal holographic recipes to an ever increasing number of model geometries and then seeking some resemblance to the real world systems without a good understanding as to why it would have to be there in the first place. In contrast, the overly pragmatic ’shut up and calculate’ approach prioritizes computational tractability over phys- ical relevance, thus making it more about the method (which readily provides a plethora of answers but may struggle to specify the pertinent questions) itself, rather than the underlying physics.

Viscosity talk in Belgrade

Today I am giving a seminar at the Institute of Physics in Belgrade.
My host is Darko Tanaskovic who has done nice work using DMFT to study bad metals and quantum criticality.

Here is the current version of the slides.
The main results are in this paper.

What is omega/T scaling?

And why is it so elusive?

Quantum many-body systems are characterised by many different energy scales (e.g. Fermi energy, Debye frequency, superconducting energy gap, Kondo temperature, ....). However, in many systems properties are "universal" in that they are determined by a single energy scale. This means that the frequency (omega) and temperature (T) dependence of a spectral function can be written in a form such as

where here  T_ K is the relevant energy scale and I set hbar =1 and k_B = 1.

However, what happens in the limit where the relevant energy scale T_K goes to zero, for example near a quantum critical point? Then the only energy scale present is that defined by the temperature T and we now expect a functional dependence of the form

This is omega/T scaling.

In one dimension the form of the scaling function is specified by conformal field theory and for quantum impurity problems (e.g. Kondo) by boundary conformal field theory.

In 1989 Varma et al. showed that many of the anomalous properties of the metallic phase of the cuprate superconductors at optimal doping could be described in terms of a “marginal Fermi liquid” self energy. They associate this with a spin (and charge) fluctuation spectrum that exhibited omega/T scaling (for all wave vectors). Specifically, the spectral function was linear in frequency at low frequencies, up to a frequency of order T.

Some claims about quantum criticality in cuprates are debatable, as discussed here.

Finding concrete realistic theoretical microscopic fermion models that exhibit such scaling has proven challenging.

In his Quantum phase transitions book Sachdev reviews several spin models (e.g. transverse field Ising model in one dimension) that exhibit omega/T scaling in the quantum critical region, associated with a quantum critical point.

 In 1999 Parcollet and Georges  considered a particular limit of a random Heisenberg model which had a spin liquid ground state and a local spin susceptibility chi’’(omega) that exhibited a form consistent with that conjectured in the marginal Fermi liquid scenario.

Local quantum criticality has been observed in a few heavy fermion compounds.  Specifically, in 2000 Schroder et al. observed inelastic neutron scattering gives the following \omega/T scaling,

In 2008 Kirchner and Si showed that near the quantum critical point in the Ising-anisotropic Bose-Fermi Kondo model (BFKM) with a sub-ohmic bath (i.e. a very specific model!) they obtained omega/T scaling similar to that associated with boundary conformal field theory, even though the model has no obvious conformal invariance.

This is my potted history and understanding. I welcome corrections and clarifications.

Quantum critical spin dynamics of a magnetic impurity in a semiconductor

There is an interesting paper
Quantum critical dynamics of a magnetic impurity in a semiconducting host
Nagamalleswararao Dasari, Swagata Acharya, A. Taraphder, Juana Moreno, Mark Jarrell, N. S. Vidhyadhiraja

The key physics of the Kondo model is the formation of a spin singlet state between the impurity spin and the spins of the electrons in the conduction band. We say, the impurity spin is “screened” by the spins in the conduction band.
The "screening" electrons involved span from the Fermi energy up to some higher energy.
The relevant energy scale is the Kondo temperature which depends in a non-analytic way on the density of states (DOS) at the Fermi energy, and is roughly the binding energy of the spin singlet.
As the DOS goes to zero the Kondo temperature goes to zero.

But, what if there is an energy gap at the Fermi energy, as in a semiconductor?
One might expect that the Kondo effect disappears and the local moment is no longer screened.
Specifically, is there a critical non-zero value of the energy gap below which the Kondo effect survives and one observes at Fermi liquid?
How about if the temperature is larger than the energy gap but less than the Kondo temperature?
Then perhaps the electrons that are thermally excited into the conduction band can screen the impurity spin.

The above fundamental questions are relevant to understanding magnetic semiconductors. They can be addressed by studying the gapped single impurity Anderson model. A number of numerical and analytical studies over the years have produced different answers to the above questions. The current paper gives definitive answers based on state-of-the art Quantum Monte Carlo calculations.

The phase diagram is shown below, with temperature versus the energy gap, delta.
Both are scaled by the Kondo temperature in the absence of the gap. LM denotes an unscreened local moment and GFL a Generalised Fermi Liquid.

The phase diagram is universal in the sense that it is independent of U in the Kondo regime (for large U) and the only relevant energy scale is the Kondo temperature (not the band width or the hybridisation energy).
It is not at all obvious (at least to me) that the universality of the delta=0 case has to extend to the non-zero delta case. But it does.

One sees that the critical value of the energy gap is zero.
Furthermore, above some non-zero temperature, of the order of a fraction of Kondo temperature and about one half of delta, a Generalised Fermi liquid forms where the local moment is completely screened.
The authors also show that the dynamic spin susceptibility associated the spin of impurity exhibits “quantum critical scaling” in the sense that it depends only on omega/T where T is the temperature and omega is the frequency.

Hopefully the paper will stimulate some experiments, either in quantum dots or in semiconductors, to observe this fascinating physics.

Comparing theory and experiment for metals: look at the frequency dependence of the reflectivity not the conductivity

The frequency dependence of the real part of the conductivity of a metal gives a lot of information, both qualitative and quantitatively. For example, one can extract a scattering rate and see if a Drude model is relevant. Hence, it is natural that experimentalists present “measurements” of this quantity.

However, it is important to acknowledge that the conductivity is not directly measured; rather, the reflectivity or absorption of a thin film or single crystal.
The real and imaginary parts of the conductivity are then extracted from a Kramers-Kronig analysis. This procedure is only stable and reliable if there is experimental data out to sufficiently high frequencies.
Several experimentalists have privately told me this can be a can of worms. It is not clear how high a frequency cutoff you need and interband transitions can complicate things…
Hence, one should be particularly nervous about people claiming exotica such as quantum criticality and non-Fermi liquid behaviour such as anomalous power laws.

There is a simple way to avoid these complications and ambiguities when comparing theory and experiment. The reflectivity can be written in terms of the conductivity as follows

From theory one can calculate the full complex conductivity and thus the reflectivity and
compare this to experiment.

This is the procedure followed by Jure Kokalj, Nigel Hussey, and I in this paper about overdoped cuprates.

I thank Swagata Acharya for motivating this post.

Violation of quantum bounds on the viscosity of strongly interacting fermion fluids

Nandan Pakhira and I just finished a paper
Shear viscosity of strongly interacting fermionic quantum fluids

Eighty years ago Eyring proposed that the shear viscosity of a liquid, η, has a quantum limit η larger than n hbar where n is the density of the fluid. Using holographic duality and the AdS/CFT correspondence in string theory Kovtun, Son, and Starinets (KSS) conjectured a universal bound η/s ≥ hbar/4πk_B for the ratio between the shear viscosity and the entropy density, s.

Using Dynamical Mean-Field Theory (DMFT) we calculate the shear viscosity and entropy density for an fermion fluid described by a single band Hubbard model at half filling. Our calculated shear viscosity as a function of temperature is compared with experimental data for liquid 3He. At low temperature the shear viscosity is found to be well above the quantum limit and is proportional to the characteristic Fermi liquid 1/T^2 dependence, where T is the temperature. With increasing temperature and interaction strength U there is significant deviation from the Fermi liquid form.

Also, the shear viscosity violates the quantum limit near the crossover from coherent quasi-particle based transport to incoherent transport (the bad metal regime). Finally, the ratio of the shear viscosity to the entropy density is found to be comparable to the KSS bound for parameters appropriate to liquid 3He. However, this bound is found to be strongly violated in the bad metal regime for parameters appropriate to lattice electronic systems such as organic charge transfer salts.

We welcome any comments.

Quantum criticality near the Mott transition in organics?

There is an interesting paper
Quantum criticality of Mott transition in organic materials 
Tetsuya Furukawa, Kazuya Miyagawa, Hiromi Taniguchi, Reizo Kato, Kazushi Kanoda

Some of the results were flagged several years ago by Kanoda in a talk at KITP.

Key to the analysis is theoretical concepts developed in three papers based on Dynamical Mean-Field Theory (DMFT) calculations

Quantum Critical Transport near the Mott Transition
by H. Terletska, J. Vučičević, Darko Tanasković, and Vlad Dobrosavljević

Finite-temperature crossover and the quantum Widom line near the Mott transition 
J. Vučičević, H. Terletska, D. Tanasković, and V. Dobrosavljević

Bad-metal behavior reveals Mott quantum criticality in doped Hubbard models
J. Vučičević, D. Tanasković, M. J. Rozenberg, and V. Dobrosavljević

The experimental authors consider three different organic charge transfer salts that undergo a metal-insulator transition as a function of pressure, with a critical point at a finite temperature. One of the phase diagrams is below.

The dots denote the Widom line determined at each temperature by the inflexion point in the resistivity versus pressure curve shown below.

They have different insulating ground states (spin liquid or antiferromagnet) and different shapes for the Widom line associated with the metal-insulator crossover above the critical temperature. Yet, the same universal behaviour is observed for the resistivity.

Aside: there is a subtle issue I raised in an earlier post. For these materials the resistivity versus temperature is non-monotonic, raising questions about what criteria you use to distinguish metals and insulators.

Here one sees that if the resistivity is scaled by the resistivity along the Widom line, then it becomes monotonic and one sees a clear distinction/bifurcation between metal and insulator.

This "collapse" of the data is very impressive.
The above universal curves then determine critical exponents through a relation

+ for insulator, - for metal
z= dynamical exponent
nu = exponent for the correlation length
The metal and insulator have the "same" temperature dependence, modulo a sign!

The data give z nu = 0.68, 0. 62, and 0.49 for the three different compounds.
This compares to the value of z nu = 0.57 obtained in the DMFT calculations, and 0.67 from a field theory from Senthil and collaborators. In contrast, Imada's "marginal theory" gives 2 and Si-MOSFETs give 1.67.

But the devil may be in the details. Some caution is in order because for me the paper raises a number of questions or concerns.

The interlayer resistivity near the critical point is about 0.1-1 Ohm-cm. Note that this about three orders of magnitude larger than the Mott-Ioffe-Regel limit.
[Aside: but caution is in order because it is very hard to accurately measure intralayer resistivity in highly anisotropic layered materials.]

Here, a rather limited temperature range is used to determine magnitude of critical exponents. The plot below shows less than a decade was used (e.g. 75-115 K). Furthermore, the authors focus on temperatures away from the critical temperature, Tc.
Ideally, critical exponents are determined over several decades and as close to the critical point as possible. The gold standard is superfluid helium in the space shuttle!

I thank Vlad for bringing the paper to my attention.

Update (26 March).
Alex Hamilton sent the following helpful comment and picture and asked me to post it.

Not that my experience in the 2D metal-insulator transition community has jaded me, but my experience is that one has to be extremely careful interpreting such 'collapses'.

1. If you have 6 orders of magnitude on the Y axis, there is no way if you can tell if an individual trace misses its neighbours by a significant margin unless there is a huge  overlap between datasets (which there usually is not) - e.g. dataset Y(1) missed Y(2) by 30% for all data points. 

2. Almost anything can be made to scale. For example consider an insulator with hopping. By definition this will show scaling behaviour. Now consider a metal with linear or quadratic in T resistance correction due to phonons. This will also fit on a scaling curve as long as the range of data in each individual dataset does not change too much on the Y-axis for the range of data in the dataset. In other words, if the overlap is not large, then I can often get scaling behaviour just by adjusting T_0 for each dataset to make them fit a common curve, because dataset Y(1) isn't that different from dataset Y(2), and neither covers an order of magnitude on the Y-axis.

Is there a quantum limit to diffusion in quantum many-body systems?

Nandan Pakhira and I recent completed a paper
Absence of a quantum limit to charge diffusion in bad metals

This work was partly motivated by

- a recent proposal, using results from the AdS-CFT correspondence, by Sean Hartnoll that there was a quantum limit to the charge diffusion constant in bad metals,

- experimental observation and theoretical calculations of a limit to the spin diffusion constant in cold atom fermions near the unitarity limit.

We calculated the temperature dependence of the charge diffusion constant in the metallic phase of a Hubbard model using Dynamical Mean-Field Theory (DMFT).

The figure below shows  the temperature dependence of the charge diffusion constant for a range of values of the Hubbard U. The temperature and energy scale is the half-bandwidth W. The Mott insulator occurs for U larger than about 3.4 W.

Violations of Hartnoll's bound occurs in the same incoherent regime as violations of the Mott-Ioffe-Regel limit on the resistivity.

We also find that the charge diffusion constant can have values orders of magnitude smaller than the cold atom bound on the spin diffusion constant.

We welcome discussion and comments.

A quantum lower bound for the charge diffusion constant in strongly correlated metals?

Previously I posted about some interesting theory and cold atom experiments that suggest that the spin diffusion constant D has a lower bound of about hbar/m, where m is the particle mass.

Coincidentally, on the same day Sean Hartnoll posted a preprint, Theory of universal incoherent metallic transport. Based on results involving holographic duality [AdS/CFT] he conjectures that the diffusion constant satisfies the bound,

D≳ℏv2F/(kBT)

where v_F is the Fermi velocity.
I have pointed out to Sean that the ratio of this lower bound for D to the cold atom one (hbar/m) is
2 T_F/T where T_F is the Fermi temperature and T the temperature. Thus, the experiments [when normalised for trap effects] and the theory give a value of D about an order of magnitude smaller than Sean's lower bound. [My earlier post also references 2D cold atom experiments that give values for D several orders of magnitude smaller].
Sean raises the issue about how much m and T_F are renormalised by interactions. However, given that the spin susceptibility undergoes a small renormalisation it is not clear to me this will be significant.
Also, in a strongly interacting system charge and spin diffusion constants might be different.

In my post I pointed out the paucity of derivations of the central equation, the "Einstein relation", D=conductivity/susceptibility. However, Sean's preprint has a nice simple derivation of this based on conservation laws, but also showing how particle-hole asymmetry complicates things.

Is publishing debatable conclusions now encouraged?

I am increasingly concerned about how many papers, particularly in luxury journals, publish claims and conclusions that appear (at least to me) to simply not follow from the data or calculations presented.
Is this problem getting worse?
Or am I just getting more sensitive about it?

Last year Nature published
Bounding the pseudogap with a line of phase transitions in YBa2Cu3O6+δ
The abstract states

Here we report that the pseudogap in YBa2Cu3O6+δ is a distinct phase, bounded by a line of phase transitions. The doping dependence of this line is such that it terminates at zero temperature inside the superconducting dome. From this we conclude that quantum criticality drives the strange metallic behaviour and therefore superconductivity in the copper oxide superconductors.

Let me examine separately the three claims I have highlighted in bold.

1. The claim that the line terminates at zero temperature is based on two data points! (the red dots in the figure below).

To be convinced of this claim I would like to see a lot more data points and particularly, extending down to a few Kelvin. Furthermore, even if you believe the authors are seeing a real continuous phase transition one wants to see that it does not terminate in a first-order line at some non-zero temperature.

1b. A line terminating at zero temperature [a quantum phase transition] is not the same as quantum criticality. Establishing that requires observing distinct features and scaling laws [e.g. a dephasing rate that is linear in temperature] at temperatures above the quantum critical point.

For the next two claims, it is important to distinguish causality and correlation. Just because one sees two things together does not mean that one causes the other. They could both be caused by some other underlying effect.

2. "quantum criticality drives the strange metal behaviour".
Here, it could be simply that in this material the phase transition and the strange metal occur at roughly the same doping. Furthermore, there are alternative explanations of the strange metal behaviour.

3. "and therefore superconductivity  in the cuprates".
I fail to see the logic. Varma's theory certainly connects quantum criticality, the strange metal, and superconductivity. But, there are alternatives that do not make this intimate connection. Hence, I can't see how it is legitimate to make this conclusion.

In contrast to the abstract, the last few sentences of the paper makes the more modest claim:

Our observed evolution of the pseudogap phase boundary from underdoped to overdosed establishes the presence of a quantum critical point inside the superconducting dome, suggesting a quantum-critical origin for both the strange metallic behaviour and the mechanism of superconducting pairing. 

Now, the story gets stranger. Look at version 1 of the paper on the arXiv. Presumably this is the version that was originally submitted to Nature. The abstract does not have the debatable sentences but instead the reasonable statements:

In slightly overdoped YBCO that transition is 20K below Tc, extending the pseudogap phase boundary inside the superconducting dome. This supports a description of the metallic state in cuprates where a pseudogap phase boundary evolves into a quantum critical point masked by the superconducting dome.

So, I would love to know whose idea it was to change the abstract? Is there any chance it was a Nature editor who wanted to "sex up" the paper?

But my real problem is not so much with this specific paper, but the many other cases I see, sometimes in non-luxury journals. Science is all about using rigorous thinking and experimentation to find out what is actually true, as best as we can tell. It is fine to speculate and to suggest possible correlations and causality. But that is totally different to claiming you have shown something to be true when you have not. We need to be precise in our language. I do think science is broken.

If we don't practice rigorous evidence-based thinking in our own community what right do we have to challenge politicians and business people who embrace climate change skepticism, opposition to vaccines, AIDS denialism, .....

Seeking definitive signatures of quantum criticality

Generally I am skeptical about quantum criticality as an important organising principle for strongly correlated electron materials (see for example, this earlier post, Are elemental metals quantum critical?). The most significant evidence is probably in heavy fermion compounds. However, particularly in the cuprates, I see quantum criticality as one of several competing "hand waiving" explainations of unusual properties. It is important to keep coming back to the idea that a true quantum critical point (QCP) will be associated with some diverging correlation length for some type of "order". Furthermore, this should lead to power laws in physical quantities over several orders in magnitude.

There is an interesting paper
Transport near a quantum critical point in BaFe2(As1−xPx)2
James G. Analytis,  H-H. Kuo, Ross D. McDonald,  Mark Wartenbe,  P. M. C. Rourke, N. E. Hussey,  and I. R. Fisher

The relevant background and context is this review article which contains the figure below

A few comments on the paper.

1. At room temperature the resistivity becomes of the order of the Mott-Ioffe-Regel limit, characteristic of a bad metal.

2. The low temperatures the metal is always a Fermi liquid [at least has a resistivity with a quadratic temperature dependence] even at dopings very close to the putative QCP. This is seen in the next figure.

3. The graph below shows the temperature dependence of the slope of the resistivity. It is striking that at high temperatures it has the approximately the same value, independent of doping. This is consistent with an earlier claim of such universality, [that even extended to elemental metals!]

4. The data above only shows a linear in T resistivity [which is often associated with quantum criticality], over less than one decade of temperature.

5. The figure below shows the doping dependence of the A coefficient of the Fermi liquid resistivity and the effective mass. Note the range of values of A is at most one order of magnitude. The effective mass only changes by about a factor of three.
In quantum critical theory both should diverge at the critical point [dashed line]. However, in the actual experimental data A does not diverge but decreases for the smallest x=0.31. This can be seen with the naked eye in the data above.

6. The colour shaded plot below shows the exponent n of the resistivity temperature dependence. The dashed line shows the Fermi liquid temperature going to zero at the putative QCP. However, as noted above even at the critical doping a Fermi liquid is always seen at low temperatures.

7. What about alternative theories?
One candidate must be Selective Mottness, as described in this preprint.
It does predict a significant enhancement doping dependence of the effective mass and the Fermi liquid coherence temperature. I am not sure how the P doping translates to band filling. Hopefully the proponents can comment.
Unlike quantum criticality Selective Mottness is actually based on calculations from a microscopic model with realistic interactions.

Belgrade bad metal talk

On thursday I am giving a seminar at the Institute of Physics in Belgrade, Serbia.
My host is Darko Tanasković. He recently did some nice work with Jaksa Vučičević, Hanna Terletska, and Vlad Dobrosavljević showing quantum critical scaling of the resistivity near the critical point of the Mott transition in Dynamical Mean-Field Theory [DMFT] of the half-filled Hubbard model. A recent PRB describes this in terms of a quantum Widom line.

Here is the current version of the slides for my talk.

In preparing the talk I realised that in some recent versions of this talk I did not includes a slide, "Open questions and future work." That is bad. Perhaps every talk should have such a slide. I want other people to work on problems I am working on and certainly don't want to create the impression that my recent work [on any topic] has "solved" the problem and there is not much left to do.

Vignettes of bad metal conference

Here is a random collection of a few of the things I learnt last week in Korea.

Yuji Matsuda described experimental work on the iron pnictides which shows evidence [via a diverging effective mass] for a Quantum critical point hidden beneath the superconducting dome.

Jan Zaanen gave an interesting talk about AdS/CFT correspondence techniques from string theory.
I am slowly becoming less skeptical about this surreal enterprise. Some concrete results that could conceivably relevant to experiment are being produced. Zaanen mentioned work by Gary Horowitz and Jorge Santos that produces a frequency-dependent conductivity that has some similarities to what is observed in the cuprates. [But, one needs to consider the alternative explanation]. I was disappointed that Zaanen ignored the work described in this post, claiming that interlayer "incoherence" in the cuprates is a mystery.

Henri Alloul recently posted on the arXiv What is the simplest model which captures the basic experimental facts of the physics of underdoped curates?
He is an experimentalist who has worked on the cuprates since the beginning. His answer is the one-band Hubbard model and Cellular Dynamical Mean-Field Theory captures the essential details.
I agree.

Alloul's text Introduction to the Physics of Electrons in Solids has been translated into English. It has a particularly nice set of problems in it.

Aharon Kapitulnik gave a talk about Polar Kerr effect as probe for time-reversal symmetry breaking (TRSB)  in unconventional superconductor. He has an instrument that can measure the magneto-optic Faraday or Kerr effects with a sensitivity of 10 nano radians! They have observed TRSB in Sr2RuO4 and the "hidden order" and superconducting states of the heavy fermion compound URu2Si2.
Non-zero Kerr effects are also seen in the pseudogap state. They can be interpreted in terms of TRSB or "gyrotopic"/chiral order associated with stripe order. [It seems the Berry phase may play a role there].

Isao Inoue, Marcelo Rozenberg and Hyuntak Kim gave talks about the emerging field of Mottronics: building transistors based on the Mott metal-insulator transition. A recent review of the field is here.

There was more, but that is enough for now...

A physical picture of spinons in two dimensions

An outstanding question in frustrated quantum magnets in two dimensions is whether one can have excitations with fractional quantum numbers, i.e. spin-1/2 spinons. [See the discussion here].

It is possible to consider triplet excitations as a pair of confined (i.e. bound) spinons. This can be worked out in detail in one dimension where the spinons are domain walls. But it has also been claimed that the spinon description is unnecessary.

There is an interesting PRL by Ying Tang and Anders Sandvik, Confinement and Deconfinement of Spinons in Two Dimensions.

They consider the spinons that are present near the quantum phase transition from a Neel ordered state to a Valence Bond State (VBS). This transition is associated with "deconfined quantum critical point" which breaks a Landau paradigm that a transition between two states that break different symmetries should be first order.

They compute the intrinsic size and the confinement length of the spinons as the quantum critical point is approached. The picture that emerges of the spinons is that of Z_4 vortices, as originally proposed in a paper by Levin and Senthil [from which the figure below is taken].

What is quantum matter?

It may depend on who you ask.
It is interesting that even twenty years ago the phrase "quantum matter" was rarely used.
Now we have

Department of Quantum Matter, Hiroshima University 

Quantum Matter Institute, University of British Columbia 

 Shoenberg Laboratory for Quantum Matter, University of Cambridge 

 So, what is quantum matter?

To some it is any material system (solid, liquid, or gas) where the quantum statistics of the constituent particles significantly affect the properties of the system. One could argue on some level this is any state of matter! After all, the Pauli exclusion principle is key to chemistry!

The above departments are largely concerned with studying what used to be called "strongly correlated electron systems". Hence, one also often sees the phrase "correlated quantum matter". I think David Pines and Piers Coleman may be two of the people who have most promoted the phrase. Coleman and Andy Schofield use the phrase "quantum matter" repeatedly in their 2005 Nature review Quantum criticality. Pines has a nice tutorial article Emergent behavior in quantum matter.
Does anyone have a better etymology?

To me the key idea is that there are states of matter [quantum many-body systems] with emergent macroscopic properties that are intrinsically quantum mechanical. Superconductivity is the classic example, being described by a macroscopic quantum mechanical wave function. Furthermore, there may not be broken symmetries. Instead, the many-body states of quantum matter may require concepts such as topological order, the most common examples being found in fractional quantum Hall effect and topological insulators. In some sense different metallic states: bad metals, "quantum critical metals", and the "strange metal" in the cuprates are all quantum matter.

The notion of quantum matter is useful as a unifying concept for describing many of the common themes of interest in two culturally distinct research communities: those studying ultracold atomic gases and correlated electron materials.

There is also a puzzling somewhat philosophical question:
Is quantum matter itself emergent or does quantum matter have emergent properties?